The Measure of Reality

Quantification and Western Society, 1250-1600

Alfred W. Crosby, Cambridge University Press, 1997

These notes provide a collage, primarily in the author's own words, of
issues and evidence in the history of quantitative literacy adapted from
"The Measure of Reality." The notes do not represent a complete or
coherent summary of the book, but merely a selection of ideas relevant to
quantitative literacy.(QL Home Page)

Europeans of the late middle ages inherited a profound change in
mentalit&eacutethat had been fermenting for centuries, a change
from the ancient qualitative way of comprehending the world--what Crosby
calls the "venerable model"--to a quantitative model that would soon
dominate Western society and provide Europe with the power to dominate the
world. Crosby's thesis is that in the late thirteenth century, beginning
around 1250, a constellation of remarkable discoveries awoke European
intellectuals and bourgeois alike to new ways of thinking--in
"quanta"--that profoundly affected the subsequent cultural and political
history of the entire world. In the space of less than a century just
before and after 1300, Europe produced its first mechanical clock (which
quantized time), marine charts and perspective painting (which quantized
space), and double-entry bookkeeping (which quantized financial
accounts).

The impact of these changes were recorded in the decades and centuries
that followed through new words and new ideas that appeared as "sparks
thrown off by the wheels of Western society grating against the edges of
old ruts." In 1300 everyone thought of nature as heterogeneous, each
quality with its own measure. Yet 250 years later Pieter Bruegel (in his
1560 painting Temperance)portrayed people engaged in visualizing
reality as aggregates of uniform units (or quanta): leagues, miles,
degrees, letters, guilders, hours, minutes, musical notes.

In the venerable model, all earthly things were ignoble and impermanent.
Left to its own devices, fire rose up toward its proper home in the sphere
of fire; stones, similarly motivated, fell straight down towards the
earth. Bruegel shows the West changing its mind, deciding to treat the
universe in terms of quanta uniform in one or more characteristics, quanta
that are often arranged in lines, squares, and other symmetrical forms
such as musical staffs, ledgers, platoons, or planetary orbits. What we
now take for granted was an entirely new mentalit&eacutein the late
middle ages.

Seeding a New Reality

The old Europeans inherited a classical world-view that they rarely
challenged. For Plato, mathematics is mental, measurement is material.
The latter "is always becoming and never is," whereas the former "always
is, and has no becoming." For Aristotle, the mathematician measures only
after he strips away sensible features such as weight, hardness, and
temperature, which the ancients did not see as quantifiable. Indeed, if
you can imagine measuring heat before the invention of a thermometer, then
why not also measure virtue, certitude, and grace? Before the invention
of appropriate measuring instruments, the former are no less ephemeral
than the latter. Intelligent people in the ancient world would not waste
time trying to measure exactly anything as variable as material reality.
The West's "distinctive intellectual accomplishment" in the late middle
ages was "to bring mathematics and measurement together."

Consider the way Europeans marked the passage of time in the early middle
ages. Because Europe did not straddle the equator, and because old
traditions dictated twelve hours for each day and each night, Europeans
developed a system of unequal "accordion-pleated hours that puffed up and
deflated" so as to ensure a dozen hours for each daytime and each
nighttime, winter and summer. Medieval Europeans were just as concerned
with time as we are, but their way had more to do with symbolic values (of
prayers) than with precision or consistency. The mechanical clock
introduced in town squares in the early fourteenth century (e.g.,
Strasbourg, 1352) revealed that "invisible, inaudible, seamless time" was
composed of quanta. Centuries later, faith in absolute time emboldened
Kepler to see that planets sweep out equal areas in equal time.

As medieval time was what happened, medieval space was what it contained.
Vacancy had no authenticity. The ranks of subjects painted were more
important than their appearances. Alberti, "an inveterate measurer,"
suggested to artists that they hang between them and their subject a thin
veil with bright threads woven to form a grid. By the fourteenth century,
painters began to paint with a picture-unit, a quantum, in mind. Some of
the rules of perspective drawing were developed by Ptolemy, whose work was
rediscovered by Europeans around 1400 in the context of depicting a curved
reality on a flat surface via a grid of lines. However, it was artists,
not cartographers, who first made good use of these rules. More than any
other method, perspective drawing satisfied the new craving for exactness
and predictability.

Another signal of the new mentalit&eacutewas the shift from aural
to visual language for important documents. Between 1220 and 1270 both
the Vatican and England's Royal Chancery increased enormously the use of
written records. In the early middle ages (500-1000) writing and reading
was laborious. Writing was just speech on a page without spaces or
punctuation. so most people read aloud. Reading then was like "walking
on stilts." The introduction of spaces and punctuation (quanta)
transformed writing and reading. By the end of the thirteenth century
reading had become silent and swift, thus more informative, private, and
"perhaps heretical." From then on, literacy--verbal or quantitative--was
visual.

Arithmetic and Mathematics

The greatest difference between medieval and renaissance thinking is in
the designation of quantity. Medieval Europeans used numbers for effect,
not for accuracy. For us, numbers are utterly neutral, free of all moral
and emotional value. Not so for the old Europeans: they thought of
numbers as qualitative and symbolic as well as quantitative. For them the
number 3 may be 1 + 1 + 1 as well as the square root of 9, but it is just
as likely to also be a reference to the Trinity. Now we use numbers
whenever we want to narrow focus and achieve precision. The old Europeans
accepted imprecision in order to grasp better what they believed to be
important--the meaning of numbers. Often they were reaching not for a
handle on reality but for a clue as to what lay beyond. Medieval
Europeans were often "as poetic about numbers as about words."

In the Middle Ages and Renaissance, numbers seethed with messages. Even
in the hands of an expert--especially in the hands of an expert--numbers
were the source of extra-quantitative news. Christian number-smiths
started down the path to mathematics as an expression of awe. Bacon, for
example employed patterns in numbers to predict the downfall of Islam.
The strong belief in the symbolic meaning of numbers undoubtedly
contributed to a lag in Europeans' development of practical
mathematics.

Hindu-Arabic numerals, algorism (strategies for calculation), place value,
and zero (cipher) appeared in Europe in the twelfth and thirteenth
centuries, but to be effective they needed to be adopted as a single
package. Reluctance to accept zero as a digit--because it was not the
count of some thing--hindered the spread of the entire system. In the
fifteenth century, businessmen found themselves drowning in a quicksand of
fractions (typical account balance: 3345312/4320864). They were rescued
by the decimal system, which itself took three hundred years to develop
(from the early thirteenth to the early sixteenth century).

The advance of practical mathematics was slowed also by the lack of clear
and simple mathematical expressions. For example, there were no universal
signs for plus, minus, equal, or root; Roman numerals prevailed in most
written records. Although the abacus (or counting board) was known in
ancient times, it mysteriously disappeared from the written and
archeological record of Europe for 500 years from 500 to 1000 A.D.
(Europeans never saw the oriental abacus; if they had, "they may never
have adopted the Hindu-Arabic numerals." They were stuck with stones
arranged on lines drawn in the sand, not nearly as efficient as beads on
wires.)

Algebraic notation remained a mishmash of words, abbreviations, and
numbers until French algebraists in the late sixteenth century began
using single letters to denote quantities--vowels for unknowns, consonants
for knowns. In the next century, Descartes shifted the tradition to
beginning and ending letters of the alphabet.

Parallel with advances in symbology was a change in the perception of the
meaning of mathematics. We now see numbers as symbols of quantities
devoid of qualities, which is why they are so useful. But the old view of
numbers as symbols of qualities was deeply entrenched. "It is simplistic
but false to believe that number mysticism retreated as practical
mathematics advanced." (Millenialists of today show that the transition
is not yet complete.)

Music, Money, and Maps

Music. Although we now think of quantification as more scientific
than artistic, it most probably first appeared in European thought through
elaboration of Gregorian chant. "Immaculately nonmensural," Gregorian
chant employs notes of arbitrary length, not exact multiples of any other
note; they are as long as they need to be. Chant provides as clear an
example of "time measured solely by its contents" as we are likely to
find.

Polyphony, which emerged in the late twelfth and early thirteenth century,
required a rhythmic measure (or quanta) to synchronize the several parts.
To help choirs learn this more complex music, choirmasters introduced
musical notes and measures (quanta), symbols for rests (silences, or
ciphers, symbols for something that is nothing), and the musical staff
(Europe's first graph). (Inexplicably, Europeans waited seven centuries
before exploiting this device to represent physical phenomena.) In
contrast to chant, notes were precise multiples or divisions of each
other. Time became the measure of sound, as well as of its omission--a
yardstick to measure things or their absence. Thus did time become
abstract. Sounds in abstract time--that is, sounds written on
paper--could be inverted, reversed, divided, and repeated. In the span of
one century the change in music from purely aural and nonmensural to
visual and quantized was so profound that a fourteenth century writer
commented about musical procedures that are "more easily seen than
heard."

Money. One of the "shattering simplifying" ideas of all time, money
quantifies everything. Around 1300 Europeans took another giant step
towards abstraction by introducing the notion of "money of account"--a
currency used for keeping books but not necessarily for actually
exchanging money. (The euro is like that now.) Money of account provided
for finance what measures did for music or minutes for time-- a common
unit of measurement. It made possible double entry bookkeeping, "a mirror
in which the adept sees both himself and others." Double entry books,
introduced at the beginning of the fourteenth century, record assets and
liabilities separately. Ever since, bookkeeping has had a pervasive
influence on the way we think, especially on our practice of dividing
things into black and white, good and evil, this or that. "In the past
seven centuries bookkeeping has done more to shape the perceptions of more
bright minds than any single innovation in philosophy or science."

Maps. In the thirteenth century maps offered not a representation
of geography but information on what was deemed to be important and
unimportant. Maps of that era were for sinners, not navigators; they were
more an expressionist portrait than a scale drawing. The idea of drawing
maps in accordance with a gridwork of lines existed in Western Europe in
the early fourteenth century, but the gridwork served only as an aid to
reproduce mariners' sketches. It took the re-entry of Ptolemy (through
his Geographia) in 1400 to treat the gridwork as coordinates on the
surface of the earth, calculated in relation to fixed stars. By 1494,
just after Columbus' discovery of the New World, Spain and Portugal mapped
boundaries in the high seas by measurements calculated by degrees. (In
practical terms, distances on water can be measured only in degrees.)

The Power of Quantification

Just how did descendents of the dark ages manage to conquer the world in a
span of just five centuries? One clue may lie in the West's flexibility.
Compared with the more advanced Arab, Indian, and Chinese civilizations,
western civilization in the middle ages lacked firmness in political,
religious, and cultural authority. Among the great civilizations, it was
unique in its stubborn resistance to centralization and standardization.
Western Europe was a warren of competing jurisdictions; no authority had
effective political, religious, or intellectual control. Thus artisans
and merchants who developed techniques that challenged the old order
quickened the climate of change. "The elites of cathedral or palace could
not suppress the merchants because they required the skills of this cocky
meritocracy."

Teachers of philosophy and theology were the most influential
intellectuals of the middle ages. They did not believe they had to invent
or discover wisdom, only rediscover it. As compilers and weavers of
approved opinion, they faced the daunting task of organizing the massive
bequests of classical, Islamic, and Christian thought. In the process of
working through contradictory texts these scholars, epitomized by Thomas
Aquinas, reinvented rigor and logic, carefully climbing ladders of
syllogisms from premise to conclusions. The next step beyond logic is
mathematics. But the fourteenth century scholars never took that step
because they did not think in terms of measured quantities. They made
great progress in mathematics by geometrizing qualities such as velocity
and temperature, but with no measurements. These scholars were
"mathematicians without being quantifiers."

According to Crosby, quantification's greatest ally is vision, a
"martinet" that encroaches on the other senses. The greatest advantage of
vision is its compatibility with measurement in terms of uniform quanta.
Visualize an idea on paper and you can divide it into equal quanta; you
can then count the quanta and think rigorously. "Record events on paper
and you have a time machine that can be read forward of backward. Record
accounts, you can work backwards to find mistakes. Record music, and you
can play the piece backwards." Written records possess an independence
that words rarely do: they can contradict your fondest wishes (e.g., as
they did to Kepler, forcing him to abandon the idealism of nested Platonic
solids for more rigorous planetary laws).

Quantification and visualization together "snap the lock" on the
rationalistic, precise, punctual character of modern culture. They
introduced to the West a faith that lasted for centuries, a faith that
mankind was capable of an intimate understanding of their universe. By
the fifteenth century, the West had a greater proportion of individuals
who understood wheels, levers, and gears than any other region on earth.
In the sixteenth century few societies equaled the West in the ability to
project power over long distances, to improvise institutions, and to
create new commercial opportunities.